LMIs in Control/pages/Discrete Time QSR

The Concept
In systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state $$ x(t) $$, its input $$u(t)$$ and its output $$y(t)$$, the input-output correlation is given a supply rate $$ w(u(t),y(t))$$. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function $$ V(x(t))$$ such that $$V(0)=0$$, $$V(x(t))\ge 0 $$ and


 * $$ \dot{V}(x(t)) \le w(u(t),y(t))$$

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate $$ w(u(t),y(t)) = u(t)^Ty(t) $$.

The physical interpretation is that $$V(x)$$ is the energy stored in the system, whereas $$w(u(t),y(t))$$ is the energy that is supplied to the system.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.

The System
Consider a discrete-time LTI system, $$\mathcal{G} : \mathcal{l}_{2e} \rightarrow \mathcal{l}_{2e}$$, with minimal state-space relization $$(\mathcal{A}_{d}, \mathcal{B}_{d}, \mathcal{C}_{d}, \mathcal{D}_{d})$$, where $$\mathcal{A}_{d} \in \mathcal{R}^{n\times n}, \mathcal{B}_{d} \in \mathcal{R}^{n\times m}, \mathcal{C}_{d} \in \mathcal{R}^{p\times n},$$ and $$\mathcal{D}_{d} \in \mathcal{R}^{p\times m} $$.


 * $$x(k+1)=\mathcal{A}_{d}x(k)+\mathcal{B}_{d}u(k)$$
 * $$y(k)=\mathcal{C}_{d}x(k)+\mathcal{D}_{d}u(k), k=0,1... $$

The Data
The matrices $$ \mathcal{A}_{d},\mathcal{B}_{d},\mathcal{C}_{d} $$ and $$\mathcal{D}_{d}$$

The Optimization Problem
The system $$\mathcal{G}$$ is QSR disipative if


 * $$\sum_{i \mathop =0}^K (\mathcal{y}_{i}^TQ\mathcal{y}_{i} + 2\mathcal{y}_{i}^TS\mathcal{u}_{i} + \mathcal{u}_{i}^TR\mathcal{u}_{i})\,dt \ge 0, \forall u \in \mathcal{l}_{2e}, \forall k \in \mathcal{Z} \ge 0 $$

where $$\mathcal{u}_{k}$$ is the input to $$\mathcal{G}, \mathcal{y}_{k}$$ is the output of $$\mathcal{G}, Q \in \mathcal{S}^{p}, S \in \mathcal{R}^{p\times m},$$ and $$\mathcal{R} \in \mathcal{S}^{m}$$.

LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems
The system $$\mathcal{G}$$ is also QSR dissipative if and only if there exists $$ P \in \mathcal{S}^{n},$$ where $$ P > 0,$$ such that


 * $$\begin{bmatrix}

A^{T}_{d}PA_{d} - P - C^{T}_{d}QC_{d} & A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d} \\ (A^{T}_{d}PB_{d} - C^{T}_{d}S - C^{T}_{d}QD_{d})^{T} &  B^{T}_{d}PB_{d} - D^{T}_{d}QD_{d} - (D^{T}_{d}S + S^{T}D_{d}) - R \end{bmatrix}\le 0.$$

Conclusion:
If there exist a positive definite $$P$$ for the the selected Q,S and R matrices then the system $$\mathcal{G}$$ is QSR dissipative.

Implementation
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs
KYP Lemma KYP Lemma for continous Time QSR Dissipative system